Implementing vector addition in Coq

Question

Implementing vector addition in some of the dependently typed languages (such as Idris) is fairly straightforward. As per the example on Wikipedia:

Answer 1

It's not possible to implement this function so easily in plain Coq: you need to rewrite your function using the convoy pattern. There was a similar question posted a while ago about this. The idea is that you need to make your match return a function in order to propagate the relation between the indices:

Set Implicit Arguments.

Inductive vector (X : Type) : nat -> Type :=
  | vnul : vector X 0
  | vcons {n : nat} (h : X) (v : vector X n) : vector X (S n).
   Arguments vnul [X].

Definition vhd (X : Type) n (v : vector X (S n)) : X :=
  match v with
  | vcons _ h _ => h
  end.

Definition vtl (X : Type) n (v : vector X (S n)) : vector X n :=
  match v with
  | vcons _ _ tl => tl
  end.

Fixpoint vpadd {n : nat} (v1 v2 : vector nat n) : vector nat n :=
  match v1 in vector _ n return vector nat n -> vector nat n with
  | vnul =>           fun _  => vnul
  | vcons _ x1 v1' => fun v2 => vcons (x1 + vhd v2) (vpadd v1' (vtl v2))
  end v2.